Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C52 · Schur multiplier (H2) = C52 |
| direct product | C4×H(3,𝔽5) |
| unambiguous semidirect product | (C4×C52)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C52 |
| commutator | C5 | C4×C52 |
| Frattini, socle, cosocle | C2×C5 | C2×C52 |
| prime radical | C2×H(3,𝔽5) | C2 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ G | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 100 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,4,8,15)(3,6,10,17,22)(5,11,20,25,21)(7,13,19,16,23)(9,18,24,14,12)(26,27,28,29);(2,5,12,13,6)(4,9,10,11,19)(8,16,20,17,18)(15,22,23,24,25)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 13);